Features of the sentence, "Every farmer who owns a donkey beats it", require careful consideration for adequate description (though reading "each" in place of "every" does simplify the formal analysis). The donkey pronoun in this case is the word it. The indefinite article 'a' is normally understood as an existential quantifier, but the most natural reading of the donkey sentence requires it to be understood as a nested universal quantifier.

There is nothing wrong with donkey sentences: they are grammatically correct, they are well-formed, their syntax is regular. They are also logically meaningful, they have well-defined truth conditions, and their semantics are unambiguous. However, it is difficult to explain how donkey sentences produce their semantic results, and how those results generalize consistently with all other language use. If such an analysis were successful, it might allow a computer program to accurately translate natural language forms into logical form.[8] The question is, how are natural language users, apparently effortlessly, agreeing on the meaning of sentences like these?

This section needs attention from an expert in logic. The specific problem is: The exposition is confused and confusing; parts of it are wrong.WikiProject Logic may be able to help recruit an expert.

Donkey sentences became a major force in advancing semantic research in the 1980s, with the introduction of discourse representation theory (DRT). During that time, an effort was made to settle the inconsistencies which arose from the attempts to translate donkey sentences into first-order logic.

Donkey sentences present the following problem, when represented in first-order logic: The systematic translation of every existential expression in the sentence into existential quantifiers produces an incorrect representation of the sentence, since it leaves a free occurrence of the variable y in BEAT(x.y):

In this case, the logical translation fails to give correct truth conditions to donkey sentences: Imagine a farmer not beating his donkey. The formula will be true in that situation, because for each farmer we need to find at least one object that either is not a donkey, or not owned by this farmer, or is beaten by the farmer. Hence, if this object denotes a pig he also owns, anything unrelated, or even the farmer himself, the sentence will be true in that situation.

A correct translation into first-order logic for the donkey sentence seems to be:

Unfortunately, this translation leads to a serious problem of inconsistency. One possible interpretation, for example, might be that every farmer that owns any donkeys beats every donkey. Clearly this is rarely the intentional meaning. Indefinites must sometimes be interpreted as existential quantifiers, and other times as universal quantifiers, without any apparent regularity.

The solution that DRT provides for the donkey sentence problem can be roughly outlined as follows: The common semantic function of non-anaphoric noun phrases is the introduction of a new discourse referent, which is in turn available for the binding of anaphoric expressions. No quantifiers are introduced into the representation, thus overcoming the scope problem that the logical translations had.

^David Lewis describes this as his motivation for considering the issue in the introduction to Papers in Philosophical Logic, a collection of reprints of his articles. "There was no satisfactory way to assign relative scopes to quantifier phrases." (CUP, 1998: 2.)